Trigonometry: From Goniometry to Triangles
This page presents a comprehensive overview of trigonometric concepts and their application to triangles. The content is organized to provide a clear understanding of various theorems and formulas essential in trigonometry.
Vocabulary: Goniometry refers to the measurement of angles and the study of angular functions.
The page begins by establishing conventions for labeling triangle elements:
- Vertices are denoted by uppercase letters
- Sides are represented by lowercase letters
- Angles are indicated using Greek letters
Definition: In trigonometry, a right-angled triangle is a triangle containing one 90-degree angle.
The document then proceeds to outline several key theorems:
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First Theorem (for right-angled triangles):
This theorem likely refers to the basic trigonometric ratios in right-angled triangles, though specific formulas are not provided in the image.
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Second Theorem:
The following formulas are presented:
- b = c tan β
- c = a cos β
- b = a cos α
Example: In a right-angled triangle, if the hypotenuse (c) is 10 units and angle β is 30°, then side b can be calculated as b = 10 * tan(30°) ≈ 5.77 units.
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Chord Theorem:
The formula a = 2r sin α is provided, where 'r' likely represents the radius of a circle and 'a' the length of a chord.
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Law of Sines:
The formula (a / sin α) = (b / sin β) = (c / sin γ) is presented, which is applicable to all triangles.
Highlight: The Law of Sines is a fundamental theorem in trigonometry, allowing for the solution of triangles when certain side lengths and angles are known.
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Consequences of the two theorems:
- The formula (side₁ * side₂) / 2 = r² * sin(included angle) is provided.
- For an inscribed angle in a semicircle: If ABC is inscribed in a semicircle with AC as the diameter, then angle ABC = 90°.
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Carnot's Theorem (Law of Cosines):
Three equivalent formulas are presented:
- b² = a² + c² - 2ac cos β
- a² = b² + c² - 2bc cos α
- c² = a² + b² - 2ab cos γ
Vocabulary: The Law of Cosines is also known as the Teorema di Carnot in Italian, named after the French mathematician Lazare Carnot.
The page concludes with a note about 'r' representing the radius of the circumscribed circle of the triangle.
This comprehensive summary covers the essential teoremi sui triangoli rettangoli and teoremi trigonometria triangoli qualsiasi, providing a solid foundation for understanding and applying trigonometric concepts to various triangle problems.